# erfc, erfcf, erfcl

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< c‎ | numeric‎ | math

Defined in header ``<math.h>`` |  |
float erfcf( float arg ); | (1) | (since C99)
double erfc( double arg ); | (2) | (since C99)
long double erfcl( long double arg ); | (3) | (since C99)
Defined in header ``<tgmath.h>`` |  |
#define erfc( arg ) | (4) | (since C99)

1-3) Computes the complementary error function of arg, that is 1.0 - erf(arg), but without loss of precision for large arg.
4) Type-generic macro: If arg has type long double, `erfcl` is called. Otherwise, if arg has integer type or the type double, `erfc` is called. Otherwise, `erfcf` is called.

### Parameters

arg | - | floating-point value

### Return value

If no errors occur, value of the complementary error function of arg, that is \(\frac{2}{\sqrt{\pi} }\int_{arg}^{\infty}{e^{-{t^2} }\mathsf{d}t}\)

2
√π

∫∞
arge-t2
dt or \({\small 1-\operatorname{erf}(arg)}\)1-erf(arg), is returned.

If a range error occurs due to underflow, the correct result (after rounding) is returned.

### Error handling

Errors are reported as specified in `math_errhandling`.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

- If the argument is +∞, +0 is returned.
- If the argument is -∞, 2 is returned.
- If the argument is NaN, NaN is returned.

### Notes

For the IEEE-compatible type `double`, underflow is guaranteed if arg > 26.55.

### Example

Run this code

    #include <math.h>
    #include <stdio.h>

    double normalCDF(double x) // Phi(-∞, x) aka N(x)
    {
        return erfc(-x / sqrt(2)) / 2;
    }

    int main(void)
    {
        puts("normal cumulative distribution function:");
        for (double n = 0; n < 1; n += 0.1)
            printf("normalCDF(%.2f) %5.2f%%\n", n, 100 * normalCDF(n));

        printf("special values:\n"
               "erfc(-Inf) = %f\n"
               "erfc(Inf) = %f\n",
               erfc(-INFINITY),
               erfc(INFINITY));
    }

Output:

    normal cumulative distribution function:
    normalCDF(0.00) 50.00%
    normalCDF(0.10) 53.98%
    normalCDF(0.20) 57.93%
    normalCDF(0.30) 61.79%
    normalCDF(0.40) 65.54%
    normalCDF(0.50) 69.15%
    normalCDF(0.60) 72.57%
    normalCDF(0.70) 75.80%
    normalCDF(0.80) 78.81%
    normalCDF(0.90) 81.59%
    normalCDF(1.00) 84.13%
    special values:
    erfc(-Inf) = 2.000000
    erfc(Inf) = 0.000000

### References

- C23 standard (ISO/IEC 9899:2024):

  - 7.12.8.2 The erfc functions (p: 249-250)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - F.10.5.2 The erfc functions (p: 525)

- C17 standard (ISO/IEC 9899:2018):

  - 7.12.8.2 The erfc functions (p: 249-250)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - F.10.5.2 The erfc functions (p: 525)

- C11 standard (ISO/IEC 9899:2011):

  - 7.12.8.2 The erfc functions (p: 249-250)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - F.10.5.2 The erfc functions (p: 525)

- C99 standard (ISO/IEC 9899:1999):

  - 7.12.8.2 The erfc functions (p: 230)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - F.9.5.2 The erfc functions (p: 462)

### See also

erferfferfl (C99)(C99)(C99) | computes error function (function)
C++ documentation for erfc

### External links

Weisstein, Eric W. "Erfc." From MathWorld — A Wolfram Web Resource.

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